Totally disconnected space: Difference between revisions
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The [[rational numbers]] form a totally disconnected space. In fact, any irrational number gives a ''disconnection'' by partitioning the rational numbers into two open subsets -- the subset of numbers less than the given irrational and the subset of numbers greater than the given irrational. | The [[rational numbers]] form a totally disconnected space. In fact, any irrational number gives a ''disconnection'' by partitioning the rational numbers into two open subsets -- the subset of numbers less than the given irrational and the subset of numbers greater than the given irrational. | ||
==References== | |||
===Textbook references=== | |||
* {{booklink|Munkres}}, Page 152, Exercise 5 (definition introduced in exercise) | |||
Revision as of 22:30, 21 April 2008
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is an opposite of connectedness
Definition
A topological space is said to be totally disconnected if its connected components are one-point sets.
Relation with other properties
Stronger properties
Examples
The rational numbers form a totally disconnected space. In fact, any irrational number gives a disconnection by partitioning the rational numbers into two open subsets -- the subset of numbers less than the given irrational and the subset of numbers greater than the given irrational.
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 152, Exercise 5 (definition introduced in exercise)